It's nice if they understand the reasons for things (hygiene, safety, chores, etc.). But it's not essential. He can choose between a bath and a shower, but he can't choose not to get clean.
When they get to be this age, they are really too big for you to force them to do anything through physically overpowering them. So you have to be smarter. They are still dependent on your for many things though, so ou do have the power even if you don't realize it. So just wait about 15 minutes until he wants something. The answer is no, not until you shower. You can send him out of the room because he smells, you can deny him certain privileges meant for older kids (like those who are old enough to take a shower), and so on. So what's important to him? Computer, TV, video games, rides to the mall, allowance, buying certain snacks, washing his favorite shirt....?? Whatever it is, you aren't in the mood for it any more than he is in the mood to get clean. These are privileges that older, mature kids EARN, not things they are entitled to. If it were my kid, I wouldn't be going out with him in an enclosed car or doing laundry that is so smelly from being worn on a dirty body. He'll run out of clothes in a week or so, right? So you teach him to do the wash. Once he gets a taste of that, he'll probably come around.
I do think a trip to the drug store for his own shampoo and deodorant/antiperspirant might be in order. Give him some choice about what products he uses just to give him a certain amount of control over the situation. If he doesn't want to do that, then give him a gift of products that will do the job. And give him his own towel and washcloth with instructions to scrub his armpits.
You can also contact the school nurse or health teacher to find out if they have already covered this information in health class or if it's planned. If he's doing it, I can guarantee other kids are, so it might be good to give the school a nudge on this topic.